Fitting Chord Into Circle
Theorem
Into a given circle, it is possible to fit a chord equal to a given line segment which is not greater than the diameter of the circle.
In the words of Euclid:
- Into a given circle to fit a straight line equal to a given straight line which is not greater than the diameter of the circle.
(The Elements: Book $\text{IV}$: Proposition $1$)
Construction
Let $ABC$ be the given circle and $D$ be the given line segment.
Let $BC$ be a diameter of circle $ABC$.
If $D = BC$ then the task is complete, as $BC$ is already fitted into $ABC$.
Otherwise, let $CE$ be cut off from $BC$ equal to $D$.
With center $C$ and radius $CE$, draw circle $EFG$.
Let $G$ be one of the points at which $EFG$ meets $ABC$.
Then $CG$ is the chord required.
Proof
As $C$ is the center of $EFG$, $CG = CE$ and so $CG = D$.
But $CG$ is a chord of $ABC$.
$\blacksquare$
Historical Note
This proof is Proposition $1$ of Book $\text{IV}$ of Euclid's The Elements.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{IV}$. Propositions